Steady State Classification of Allee Effect System

TL;DR

The authors tackle steady-state classification for a multi-patch Allee effect system by proving a dimension-reduction result that confines steady-state coordinates to at most three distinct values, and by introducing a border-polynomial framework to partition parameter space into regions with a fixed number of steady states. They derive explicit two- and three-number reduced systems, construct general border polynomials for arbitrary dimension, and present an algorithm that combines border-polynomial continuity with CAD to count steady states. The approach yields complete classifications for $n=4$–$7$ and demonstrates computational feasibility up to larger $n$, outperforming standard CAD-based methods in efficiency. This work advances multistationarity analysis in coupled ecological models by providing scalable, algebraic tools for parameter-region classification and state-count determination.

Abstract

In this paper, we consider the steady state classification problem of the Allee effect system for multiple tribes. First, we reduce the high-dimensional model into several two-dimensional and three-dimensional algebraic systems such that we can prove a comprehensive formula of the border polynomial for arbitrary dimension. Then, we propose an efficient algorithm for classifying the generic parameters according to the number of steady states, and we successfully complete the computation for up to the seven-dimensional Allee effect system.

Figures (11)

• Figure 1: $g(z)=K$ has at most three real solutions.
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Theorems & Definitions (13)

• Theorem 2
• proof
• Definition 1
• Remark 1
• Remark 2
• Theorem 3
• Remark 3
• Example 1
• Remark 4
• Remark 5